OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A MULTI-OBJECTIVE TRANSPORTATION PROBLEM

Transportation problem is an integral part of this competitive world. Minimizing transportation cost of a product from sources to destinations is the main objective of traditional transportation problems. In the present context single objective transportation problem fails to meet the managerial decision making requirements which demands the multi objective transportation problems. Many real life problems can be formulated as multi objective transportation problem, and their solution requires consideration of conflicting objectives. Naturally, these problems have a number of potentially Pareto-optimal solutions. Multi-objective transportation problem refers to a special class of vector minimum linear programming problem, which usually involves multiple, conflicting, and incommensurable objective functions. Solving these complexities, the decision maker has to develop a utility function. In this paper, we develop a utility function to attain the targeted goal for a multi objective transportation problem. This utility function is developed using fuzzy membership functions considering deviations of objective function from the goal value where deviation variable has chosen as fuzzy type. Moreover, fuzzy max-min operator has been included for taking limiting value of the objective functions. Performance of the utility function to achieve proper goal is evaluated and justified by solving numerical examples using fuzzy goal programming.

The real world conditions of transportation pose new challenges for development of intelligent assistive systems for effective decision making. The ignorance of relationships among various attributes and parameters is one of the major causes of uncertainty in existing models of transportation problems. Fuzzy sets have been explored by various investigators to address the uncertainty issues due to relationships among various attributes. However fuzzy sets are not fully capable of dealing all kinds of uncertainty to provide intelligent solutions of transportation problems. A soft set is explored in this paper to model the uncertainty arising from the relationships of attributes with the parameters in a multimodal and multi objective transportation problem. The three modes of transportation incorporated in this model are road, rail and air. The real data set is prepared using the tariff, distance and time duration of transport available on websites of transport service agencies. The objectives are to minimize the cost and time duration of transportation. The proposed model evaluates multiple criteria using various combinations of different modes of transport to optimize the objectives of the problem. The model has been illustrated using the real data set using existing methods of solution. The soft set approach is found to be quite effective in dealing with the relationships of attributes with the parameters arising due to multi criteria decision making and leads to intelligent optimal solutions. The proposed model can be used as intelligent assistive system for decision making of multi objective and multimodal transportation problems.

This paper explores the study of multi-choice multi-objective transportation problem (MCMTP) under the environment of utility function approach. MCMTP is converted to multi-objective transportation problems (MOTP) by transforming the multi-choice parameters like cost, demand, and supply to real-valued parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTP are solved by goal programming (GP) approach. Using GP, the solution of MOTP may not be satisfied all the time by the decision maker (DM) when the proposed problem contains interval-valued aspiration level. To overcome this difficulty, here we propose the approaches of revised multi-choice goal programming (RMCGP) and utility function into the MOTP and then compared the solution between them. Finally, numerical examples are presented to show the feasibility and usefulness of our paper.

In classical fractional transportation programming problem, we want to optimize the objective function in the form of one or several ratios subject to some linear constraints. If in multi-objective transportation problem, objective function is in ration of two linear function under some linear restrictions, then the problem is called multi-objective linear fractional transportation problem (MOLFTP). In this paper we propose a new method to solve multi-objective linear fractional transportation problem which is extension of Nomani et al. (Int J Manag Sci Eng Manag (2016) [9]). Two numerical problems are presented to validate the proposed algorithm.

This chapter presents solution procedures for solving unbalanced multi-objective multi-choice stochastic transportation problems in a hybrid fuzzy uncertain environment. In this chapter, various types of unbalanced multi-objective fuzzy stochastic transportation models are considered with the assumption that the parameters representing supplies of the products at the origins and demands of the products at the destinations, capacity of the conveyances, associated with the system constraints are either fuzzy numbers (FNs) or fuzzy random variables (FRVs) with some known continuous fuzzy probability distributions. The multi-choice cost parameters are considered as FNs. In this chapter, two objectives are considered: total transportation cost and total transportation time. As the transportation cost mainly depends on fuel prices and since fuel prices are highly fluctuating, the cost parameters are taken as multi-choice cost parameters with possibilistic uncertain nature. The time of transportation mainly depends on vehicle conditions, quality of roads, and road congestion. Due to these uncertain natures, the parameters representing time of transportation are also taken as fuzzy uncertain multi-choice parameters. In this transportation model, these objectives are minimized satisfying the constraints: product availability constraints, requirement of the product constraints, and capacity of the conveyance constraints. Numerical examples are provided for the sake of illustration of the methodology presented in this chapter, and also achieved solutions are compared with the solutions obtained by some existing methodologies to establish its effectiveness.

In this problems is shown that the problem of allocating units of blood from the central depots of a regional blood transfusion centre to the hospital blood banks of its area, can be formulated as a multiobjective transportation problem. Various objectives can be taken into consideration such as the minimisation of the expected expiries, the minimisation of the variance of the expiries, transportation costs or distances etc.. An application of the model is examined.

Disasters are the unfortunate suddenly occurring events which collapse a living society within a moment. Facebook, the most trending social media have initiated an active role to response the crisis developing a safety check feature to know about the demand of victims in an area. Through this feature the Facebook authority can confirm the requirement of the affected people in a specific area. This research have introduced a mathematical model for multi-objective solid transportation problem to quick delivery of the relief products to the devastating society according to the most prior demand of an area acquiring the best information through safety check feature launched by Facebook with limited resources. The two main objectives of the model are to minimize the total cost and minimize the total time so that relief can be executed under a limited budget responding quickly to the affected area. In this paper, fuzzy goal programming is adopted to solve the multi-objective solid transportation problem. To contemplate the performance of the model, a numerical example is presented in this paper. The numerical example is solved with LINGO optimization solver.

The single-objective fuzzy or non-fuzzy transportation problem (TP) is not capable of dealing with real-life decision-making problems due to our current competitive market state. In this article, we investigate a fully integer interval multi-objective transportation problem (FIIMOTP) and a fully fuzzy integer multi-objective transportation problem (FFIMOTP). We also provide two solution approaches for solving the FIIMOTP and FFIMOTP. Numerical examples are provided to validate these two approaches. Our results show that the proposed algorithm hugely outperforms the best solution approaches.

In this paper, a new compromise algorithm for multi-objective transportation problem (MO-TP) is developed, which is inspired by Zimmermann's fuzzy programming and the neutrosophic set terminology. The proposed NCPA is characterized by assigning three membership functions for each objective namely, truth membership, indeterminacy membership and falsity membership. With the membership functions for all objectives, a neutrosophic compromise programming model is constructed with the aim to find best compromise solution (BCS). This model can cover a wide spectrum of BCSs by controlling the membership functions interactively. The performance of the NCPA is validated by measuring the ranking degree using TOPSIS approach. Illustrative examples are reported and compared with exists models in the literature. Based on the provided comparisons, NCPA is superior to fuzzy and different approaches.

Multi-objective optimization model for railway heavy-haul traffic: Addressing carbon emissions reduction and transport efficiency improvement

In the present paper, a fuzzy programming model with quadratic membership functions has been developed for the solution of a Multi-Objective Transportation problem. In literature, several fuzzy programming approaches exist with various types of membership functions such as linear, exponential, hyperbolic etc. These membership functions are defined, by taking the lower and upper values of the objective functions into account. In some cases, these methods fail to obtain an integer compromise optimal solution. In the present method, two coefficients of the quadratic membership functions are determined by the lower and upper values of the objective functions. The other coefficient is taken as a variable in the fuzzy programming approach. This means that the membership curve is fixed at the two end points and set free in between. Application of the method on numerical examples proved that the approach could generate integer compromise optimal solutions.

DINAS (Dynamic Interactive Network Analysis System) is an interactive system to aid in the solution of various multiobjective transportation problems with facility location. DINAS utilizes an extension of the reference point approach for interactive handling multiple objectives. In this approach the decision-maker forms his requirements in terms of aspiration and reservation levels, i.e., he specifies acceptable and required values for given objectives. A special solver was developed to provide DINAS with solutions to single-objective problems. It is based on the branch and bound scheme with a pioneering implementation of the simplex special ordered network (SON) algorithm with implicit representation of the simple and variable upper bounds (SUB & VUB). DINAS is prepared for IBM-PC XT/AT or compatibles as a menu-driven and easy in usage system equipped with a special network editor which reduces to minimum effort associated with data input for real-life problems.

A multi-objective transportation problem with cost dependent credit period policy under Gaussian fuzzy environment

Preference modelling by estimating local utility functions for multiobjective optimization

The main proposal of this paper is to derive two different reduction process for a trapezoidal type-2 fuzzy number. The first reduction method is based on critical values and the second method is based on α-cut of fuzzy number. As an application a multi-objective solid transportation problem, minimizing the cost and time has been developed using trapezoidal type-2 fuzzy number as system parameters and hereby solved. Finally after solving the proposed multi-objective problem by intuitionistic fuzzy programming technique a comparison between the two proposed reduction methods are discussed briefly. The proposed models and techniques are finally illustrated by providing numerical examples at the end. Also this paper present a comparative study between the proposed method to the KM algorithm and NT method for type reduction.